to my own surprise the 10,75 x 57 and the 10,75x61 have more or less the same oal

that means they have more or less the same case capacity and would be ballistically very similar.
why does the 10,75x61 exist than? I believe only because someone thought the Haenel cartridge was the better design and order this.
its like the 400/360 Purdey and the 400/360 Westley Richards, two cartridges dying out and the better 9,3x74R survive. I will ask my friend again for the oal of his round to confirm it.
The 10,75 was imho the older cartridge maybe invented around the year 1900 when the 8x57 cartridge M 88 was necked down to the 6,5x57, than necked up to 9x57, 10,75x57 and 9,3x57.

the 10,75x57 was loaded with 3,5 (53,9 grains) gramm Rottweil R 5 powder and the 350 grains bullet giving a V0 of 620 m/sec.
R 5 was a powder that seems to be very similar to Vithavuori N 140, look here for similar powders to get an idea what to do:

thats the cartridge of my friend

and that is a detail from the DWM case book.I am not authorized to show the whole page.
this book was hand written by the DWM staff over the decades. every cartridge that was notice by DWM was written in this book made later or not, old or new. there it got the DWM case number that was used later itf the case came into production

its called there "Pirschbüchse Kaliber 10,5 mm Haenel/Suhl" Pirschbüchse means stalking rifle. why was it written as caliber 10,5 mm? we don't know because its indeed the 10,75.